Question

1) Nucor a manufacturing company has received orders for 80 metal lockers, 75 desks, 200 Shelves and 25 TV stands for the month of August. Nucor sells metal lockers for $90 each, shelves for $150 each, TV stands for $65 each and desks for $95 each.

Nucor has the option of manufacturing these products in house or purchasing them from an outside supplier. The associated costs are given below

Product	Cost if Manufactured	Cost if purchased
Locker	75	80
Shelf	125	140
TV stand	30	45
Desk	48	62

The company has three types of welding machines: 10 type P, 6 type Q and 5 type R.

Type P machines can complete 5 jobs/month, Q machines can complete 10 jobs/month and R machines can complete 11 jobs/month. Machine type Q and R can be used to weld lockers and desks. Machine type P and Q can be used to weld shelves and TV stands.

The number of lockers purchased cannot exceed the number of lockers manufactures

The number of desks manufactured must be no more than twice the number of shelves manufactured

At least 30% of the TV stands must be manufactured.

The total number of goods purchased must be within 25% of the total number of goods manufactured.

Formulate a Linear Programming model that will maximize profit for Nucor.

Answer 1

LP Model:

Product	Monthly Demand	Cost if Manufactured	Cost if purchased	Selling price
Locker	80	$75	$80	$90
Shelf	200	$125	$140	$150
TV stand	25	$30	$45	$65
Desk	75	$48	$62	$95
Welding Machine
type	no of m/c	jobs/month/mc	tota job/month
P	10	5	50	shelves and TV stands
Q	6	10	60	shelves, TV stands, lockers and desks
R	5	11	55	lockers and desks
Decision Variables:
Let l1,s1,t1 and d1 represent the no of lockers, shelves, TV stands and desks manufactured
Let l2,s2,t2 and d2 represent the no of lockers, shelves, TV stands and desks purchased.
Constraints:
l1,s1,t1,d1,l2,s2,t2,d2>=0, int
l2<=l1	number of lockers purchased cannot exceed the number of lockers manufactures
d1<=2*s1	number of desks manufactured must be no more than twice the number of shelves manufactured
t1>=30%*(t1+t2)	At least 30% of the TV stands must be manufactured.
l2+s2+t2+d2<=25%(l1+s1+t1+d1)	total number of goods purchased must be within 25% of the total number of goods manufactured.
l1+l2>=80	demand
s1+s2>=200	demand
t1+t2>=25	demand
sd1+d2>=75	demand
s1+t1<=(50+60)	machine capacity
l1+d1<=(60+55)	machine capacity
l1+s1+t1+d1<=(50+60+55)	total machine capacity
Objective Function:
Maximize Profit
Profit = Sales-Cost
Sales = (l1+l2)*90+(s1+s2)*150+(t1+t2)*65+(d1+d2)*95
Cost = 75l1+80l2+125s1+140s2+30t1+45t2+48d1+62d2

Model in Excel:

Solver solution: